On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes

It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n²). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyzing what occurs if the points are chosen from a 2-dimensional region in 3-dimensional space. As an example, we examine the situation when the points are drawn from a Poisson distribution with rate n on the surface of a convex polytope. We prove that, in this case, the expected complexity of the resulting Voronoi diagram is O(n).     

Hydroxide ion initiated reactions under phase transfer catalysis conditions II. The roles of water and quat

The effects of water molecules and quat structure are shown to be significant in determining the behavior of alkylation reactions of weakly acidic carbon acids under PTC/OH^? conditions.     

An insight into the aromaticity of fullerene anions: experimental evidence for diamagnetic ring currents in the five-membered rings of C(60)(6-) and C(70)(6-)

Reduction of the two "closed" [6,6] methanofullerenes, [6,6]C(61)H(2) (1) and [6,6]C(71)H(2) (5), to the corresponding hexaanions with lithium metal causes the bridgehead-bridgehead bonds to open, at least partially, and this change gives rise to diamagnetic ring currents in the resulting homoconjugated six-membered rings (6-MRs). These new ring currents shield the overlying hydrogen atoms on the methylene bridge and induce upfield shifts of 1.60 and 0.11 ppm in their (1)H NMR resonances, respectively. Analogous reduction of the already "open" [5,6]methanofullerenes, [5,6]C(61)H(2) (2) and [5,6]C(71)H(2) (3 and 4), only slightly enhances the shielding of the hydrogen atoms over the homoconjugated 6-MRs (upfield shifts of 0.13, 0.68, and 0.14 ppm, respectively) but leads to exceptionally strong diamagnetic ring currents in the homoconjugated five- membered rings (5-MRs), as evidenced by dramatic shielding of the hydrogen atoms situated over them (upfield shift of 5.01, 6.78, and 1.63 ppm, respectively). The strongest shielding is seen for the hydrogen atom sitting over the 5-MR at the pole of C(71)H(2)(6)(-) (delta = -0.255 ppm) indicating that the excess charge density is concentrated at the poles.     

1H NMR isotope shifts arising from the substitution of 12C by 13C: application to polycyclic aromatic hydrocarbon anions

Isotope shifts are a well-established tool for structural analysis by NMR. The substitution of a protium with a deuterium is the most widely studied of these effects. However, such studies call for specific deuteration that requires complex synthetic techniques owing to the low natural abundance of deuterium. ¹³C occurs at a higher natural abundance and couples strongly with its attached proton. We have developed and refined a method to reliably observe these much smaller shifts without needing to resort to chemical synthesis. We show that carbon induced isotope shifts reflect structural features. Copyright © 2001 John Wiley & Sons, Ltd.     

Two helium atoms inside fullerenes: probing the internal magnetic field in C60(6-) and C70(6-)

A 3He NMR resonance of C606- containing He is assigned to He2@C606-, thus showing that C60 can also accommodate two helium atoms. The ratio of the di-helium compound relative to the mono- is 1:200, 10 times lower than the equivalent counterpart of C70. The 3He NMR chemical shift of He2@C606- is 0.093 ppm downfield from the already known resonance of He@C606-. In the reduced endohedral mono- and di-helium C70, the 3He NMR chemical shift of He2@C706- is 0.154 ppm upfield from the peak of He@C706-.     

Paratropicity and antiaromaticity: role of the homo-lumo energy gap

Doubly charged systems derived from fused benzenoid polycycles reveal an unquenched delocalization of 4 n π-electrons and hence are predicted to possess antiaromatic character. The magnitude of the paratropic ¹H NMR chemical shifts, due solely to the paramagnetic secondary field sustained in these species, was found to depend linearly upon the magnitude of LUMO-HOMO energy gaps of the corresponding systems. The existence of such a correlation enables a comprehensive treatment of the various factors which determine the antiaromatic character and the subtle interrelations between those factors. This, in turn, leads to a deeper understanding of antiaromaticity.     

r-convex functions

A family of real functions, calledr-convex functions, which represents a generalization of the notion of convexity is introduced. This family properly includes the family of convex functions and is included in the family of quasiconvex functions. Some properties ofr-convex functions are derived and relations with other generalizations of convex functions are discussed.Portions of this paper were presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.Research for this paper was supported in part by the Gerard Swope Fund at the Technion.     

Coverings of transfinite matrices

Subsets of a Cartesian product X × Y, where X and Y are arbitrary sets, are considered as a generalization of incidence matrices. Minimal cover, essential set etc. are introduced in a stronger sense and their properties discussed. The existence of a minimal cover for an arbitrary generalized incidence matrix is proved. As an application a previous result is extended.     

Design of insurance contracts using stochastic programming in forestry planning

This work addresses a tactical planning problem faced by a forestry firm, deciding which timber units to harvest and what roads to build to obtain the greatest possible benefits. We include uncertainty in prices by means of utility theory. This enables solutions to be found that the firm finds preferable to those obtained when risk aversion is ignored and makes it possible to design insurance contracts that benefit the firm while also being attractive to an insurer. Two types of contract are designed; one dependent on the firm’s operating result and the other independent of it. Metrics are then developed to quantify the benefits conferred by a contract, demonstrating that the latter contract type dominates the former. These results are then illustrated by applying them to a simplified planning problem of a forest owned by the Chilean forestry operator Millalemu.